902 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



We assume that D„, and dDJd^ have no common zero other than /8 = 0, so 

 that (10) gives a solution of the problem (1) for the exceptional case {cf. Art. 30, 

 at the end). 



Since Pq, ^^o, ^q are arbitrary, we may in (9) put any two of them equal to zero. 

 Thus, if /3 is a zero of D„„ other than (3 = 0, the solutions 



{'•;■> {r-cf") "'■ {\:-) - c^'tf") }.'.-^= :■:'«(«-»■) . ^ (.., 



give vanishing values of pp, pw, and pz at the surface of the cylinder. 

 Since D,;i= 0, we have, of course, 



Ai:Bi:Ci = A2:B2: €2 = 13:63:03 (12) 



so that the three forms in (11) are really one and the same solution, except for a 

 constant multiplier. This solution will be called a transitory free mode of equilibrium 

 of the cylinder. 



For each value of m the function D„j has an infinite number of zeroes, so that the 

 fundamental numbers /3 form a doubly infinite system. 



For every m an infinite number of the zeroes of D,„ are real, and an infinite number 

 complex, but none are purely imaginary (Art. 30). 



Since /3~^"'D„i is an even function of ^, the zeroes occur in pairs differing 

 only in sign. 



In (2) and (8) the expansions of ^, 0, and \|^ in ascending powers of /3 contain 

 terms of negative degree in /S, the coefficients of these being rational integral harmonic 

 functions of x, y, z. If the coefficients of a negative power (S"" be taken as the 

 ^, 0, and ^ defining a strain, these can give no traction at jO = a, since the coefficient 

 of i^"" is zero in the expansions of the values (1) of the surface tractions. 



When m>l, the displacements deduced from such values of (^, 0, -^ vanish identi- 

 cally ; * but when n% = 1 and ra == 0, they give solutions under no body force or surface 

 traction which we call the permanent free modes of equilibrium of the cylinder. Of 

 these it is convenient to consider that there are twelve, viz. six permanent st^-aining 

 modes, and the six rigid body dis2')lacements. The former are the solutions which 

 were obtained synthetically by St-Venant. 



The permanent and transitory free modes together constitute a system of funda- 

 Tnental solutions in terms of which, as will be shown, any deformation of the cylinder 

 may be expressed, within a region where there is no applied force. 



4. Statement of the general problem, of surf ace traction. Fourier s TJteorems. 

 We require values of <p, 0, \|^ which shall give, in accordance with 2 (8), at p = a, 



the normal traction pp = N(a), z), ^ 



the transverse traction pw— T(a), 2), 1- ■ • • • • (0 



the longitudinal traction pz= L(a), z), J 



* Cf. Art. 11. 



